CHAOS
VOLUME 10, NUMBER 1
MARCH 2000
Eulerian mean flow from an instability of convective plumes
Stephen Childressa)
Applied Mathematics Laboratory, Courant Institute of Mathematical Sciences, 251 Mercer Street,
New York, New York 10012
共Received 6 May 1999; accepted for publication 30 July 1999兲
The dynamical origin of large-scale flows in systems driven by concentrated Archimedean forces is
considered. A two-dimensional model of plumes, such as those observed in thermal convection at
large Rayleigh and Prandtl numbers, is introduced. From this model, we deduce the onset of mean
flow as an instability of a convective state consisting of parallel vertical flow supported by buoyancy
forces. The form of the linear equation governing the instability is derived and two modes of
instability are discussed, one of which leads to the onset of steady Eulerian mean flow in the system.
We are thus able to link the origin of mean flow precisely to the profiles of the unperturbed plumes.
The form of the nonlinear partial differential equation governing the Eulerian mean flow, including
nonlinear effects, is derived in one special case. The extension to three dimensions is outlined.
© 2000 American Institute of Physics. 关S1054-1500共00兲01101-0兴
Thus our model is focused on the ‘‘central region’’ of flow,
and does not consider the interaction of the basic state with
the walls. We are then able to crudely model perturbations of
slender plumes, but not the processes by which the plumes
are created, or the results of collisions of plumes with a
boundary. We shall allow, however, for the breaking of top–
bottom 共TB兲 symmetry. An example of this symmetry breaking is thermal convection between constant temperature
walls, but with a fluid whose viscosity depends upon temperature. In an isoviscous fluid the symmetry can be broken
by the boundary conditions, e.g., by introducing salt water
into fresh at one wall with a compensating fresh water flow
at the opposite wall, or by varying the temperature horizontally on one wall. Since we shall be concerned only with the
central region, we adopt the simplest isoviscous model but
allow the basic state to break TB symmetry.
Krishnamurti and Howard1 first observed mean flows in
classical isoviscous thermal convection, as a component of a
convective state consisting of tilted rolls. They also found
that the mean Reynolds stresses driving the flow were balanced by viscous stresses of the mean flow in the steady
state, and noted the distinction between Eulerian mean flow,
which can involve no net mean transport of a given fluid
parcel, and Lagrangian mean flow, where, by definition, particle transport occurs.
In order to understand the origin of a mean flow in their
experiment, Howard and Krishnamurti2 considered a truncated system of equations, which enlarged the Lorenz model
of convection to allow a mean-flow mode. In a thorough
study of the bifurcation structure of the resulting fivedimensional system, they confirmed the origin of the Eulerian mean flow as a secondary bifurcation from the basic
conductive state, which accompanies the onset of tilting of
steady rolls created by the basic Bénard instability. A tertiary
bifurcation results in unsteady tilted rolls and the onset of
Lagrangian mean flow. Tilted rolls can be viewed as precursors of the fully developed tilted plumes we study below.
In 2D with flow in the x⫺z plane, z upward, tilted rolls
While mean flows are observed in numerous flows driven
by buoyancy forces, the origin of the stresses needed to
set up such flows is not well understood, particularly
when the system is well beyond the onset of convective
instability. In this paper we treat the generation of mean
flow by discrete, thin plumes. Such plumes are often observed in convection at high Rayleigh numbers. To simplify our analysis we focus on two-dimensional flow, and
study the stability of a periodic array of parallel plumes.
Two modes of instability are deduced, one of which is
associated with an Eulerian mean flow. By introducing a
boundary-value problem that isolates the latter mode of
instability, we treat higher-order terms in an expansions
and derive an equation for the evolution of the mean
flow. The extension of the method to three dimensions is
discussed.
I. INTRODUCTION
It is now well-established that a large-scale or mean
shear flow is a natural accompaniment of many systems
driven by buoyancy or Archimedean forces. Mean flows, in
either the Eulerian or Lagrangian sense, can be important
contributors to transport, and so their origin in flows driven
by Archimedean forces is of practical importance. Here we
understand an Archimedean force to be a density variation
caused by a single scalar field such as temperature or salt
concentration. In this paper we are concerned with the dynamical processes driving these mean flows. We shall study
this problem in a very simple two-dimensional thermal
model in which the basic unperturbed state is a parallel flow
driven by a horizontal gradient of temperature. We further
assume that the horizontal scale of this basic state is small
compared to some vertical scale, presumably the distance
between two horizontal walls containing the fluid. We shall
not, however, explicitly consider the flow near these walls.
a兲
Electronic mail: childres@math3.nyu.edu
1054-1500/2000/10(1)/28/11/$17.00
28
© 2000 American Institute of Physics
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Chaos, Vol. 10, No. 1, 2000
create a nonzero horizontal average of uw, leading to a force
which is minus the z-derivative of the latter quantity. This
force is in equilibrium with the viscous force of the mean
flow, here the horizontal mean of u. Many subsequent papers
have examined various aspects of mean flow structure, using
low-order ODE models or full numerical simulation in convection and magnetoconvection see, e.g., Refs. 3–7,14,15.
Related Archimedean instabilities, associated with larger
scales of motion, occur in doubly diffusive systems as instabilities of salt fingers; see, e.g., Ref. 6. The linear stability of
these fingers, with negligible diffusivity of the salt, has been
discussed in Ref. 7. There a vertically uniform unperturbed
state similar to that considered below was found to be unstable, and in certain cases the mode of instability involved
horizontal mean flow.
For fluids with variable viscosity, Solomatov8 divided
the thermal convective states into three regimes. The first is
essentially isoviscous, with upwelling plume and downwelling plumes of roughly the same properties. When the
contrast is in the range 104 ⫺105 plumes of both types coexist, but are different in structure. For still higher contrasts a
‘‘stagnant lid’’ regime is reached, where essentially isoviscous convection by upwelling plumes occurs below a lid of
very viscous fluid, where the transport is mainly by conduction. Recent computations have gone as high as 105 共Ref. 9兲
and 106 共Ref. 10兲, and observed strong mean flows at Rayleigh numbers up to 104 .
The present paper was motivated by recent experiments
in glycerol, a common working fluid with a significant dependence of viscosity upon temperature.11,12 In these experiments the mean center 共or interior兲 temperature was measured and a mean flow observed in an 18 cm cube at
Rayleigh numbers 106 ⫺109 and Prandtl numbers 102
⫺103 , with a modest viscosity contrast ⬃200. A shadowgraph of the plume structure, clearly showing a mean flow
共in fact, a cellular flow across the diagonal plane兲 is shown in
plate I.
Instability of convective plumes
29
ture, well outside the thin thermal boundary layers on each
wall, was deduced from a boundary-layer analysis of the wall
region.
These experiments suggested that an analysis of the onset of mean flow in fully-developed convection might be
possible by taking slender plumes as the underlying steady
flow, in the place of marginally convective Bénard cells. The
relative ease of setting up the mean flow in the presence of
top–bottom asymmetry suggested a parallel investigation of
the influence of TB symmetry on the Reynolds stresses responsible for the mean flow. In such an approach marginal
stability can still be investigated, but about a state quite different from that of marginal convection.1 Even at high Rayleigh numbers, it is observed that convective flows may or
may not drive mean flows, so a criterion based upon the
properties of plumes is of some interest.
There seems to be little hope of finding useful exact
solutions of the nonlinear equations, and the simplified
model proposed below, where wall effects are neglected, becomes attractive. We shall formulate a 2D model and discuss
the unperturbed plumes in Sec. II. In Sec. III a basic invariance of slender plumes associated with a horizontal shift of
their positions is discussed, and shown to be unrelated to the
generation of an Eulerian mean flow. In Sec. IV the meanflow instability is identified as a degenerate bifurcation of the
solutions of the linear stability problem, at a zero eigenvalue
of geometric multiplicity two. In Sec. V the form of the
linear mean-flow equation is derived, and the mechanism is
shown by an example to lead to instability in a case with TB
symmetry.
Some properties of the partial differential equation governing the mean flow are given in Sec. V, including the form
of the nonlinear terms and the role of breaking the TB symmetry. Finally, in Sec. VI we outline an extension of the
theory to three dimensions.
We shall deal in this paper only with the onset of Eulerian mean flow. This is analogous to studying the addition of
a parallel flow 关 U(z),0兴 to the two-dimensional cellular flow
of the form (u,w)⫽( ␺ z ,⫺ ␺ x ), ␺ ⫽sin(ax)sin(by), where,
since thin plumes are considered, we would take bⰆa. The
mean flow causes the ‘‘channels’’ to develop between the
cells, where transport of scalar and vector fields is enhanced;
see, e.g., Ref. 13. In a real flow, plume creation is a random
process and transport would be chaotic in both space and
time. Our purpose in the present investigation is to indicate
how mean flow is generated in the simplest periodic geometry, with the understanding that the same dynamical effects
would apply for irregular patterns. The averaging process
needed in that case is discussed briefly in Sec. VI.
II. FORMULATION OF THE MODEL
Note that there is a small but noticeable difference in the
boundary-layer structure and the plume geometry of the top
and bottom walls. The thickness of the thermal layers differed by a factor ⬃3 in these experiments. In Ref. 11, a
model allowing calculation of the average center tempera-
We shall deal here with thermal convection between
horizontal planes z⫽0,H. We consider the customary equations of thermal convection of a Boussinesq fluid of constant
viscosity. In a fluid of constant viscosity and isothermal
walls, the Boussinesq approximation leads to equations
which possess the top–bottom symmetry mentioned above.
Indeed, the equations are invariant under a shift of origin of
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30
Chaos, Vol. 10, No. 1, 2000
Stephen Childress
the temperature scale, so we may assume that the wall temperatures satisfy T 2 ⫽⫺T 1 . Then the equation and boundary
conditions are invariant under the transformation z,w,T
→⫺z,⫺w,⫺T where w is the vertical velocity component.
In such a fluid, regardless of the Rayleigh and Prandtl numbers, the convective fields associated with the top and bottom
walls are effectively indistinguishable. As mentioned in the
Introduction, we shall here allow top–bottom symmetry to
be broken, although this will be introduced through the
choice of the unperturbed state of model flow. We shall do
this without reference to whatever boundary conditions at the
wall might cause the broken symmetry; see Sec. II B. 关The
presence of the walls will appear in one way only, as a condition of zero net vertical mass flux, see 共15兲.兴
The equations of motion are then
共1兲
⳵ t T⫹q"“T⫺ ␬ T ⵜ 2 T⫽0,
共2兲
“"q⫽0.
共3兲
All symbols are conventional, ␣ being the coefficient of thermal expansion of the fluid and ␬ T the thermal diffusivity.
A. Dimensionless variables
We now convert 共1兲–共3兲 to suitable dimensionless form.
We shall be dealing with temperature fields whose horizontal
scale of variation, measured by a length L, is small compared
to H, L/HⰆ1. At the same time we shall exploit a high
Prandtl number limit to neglect the diffusion of temperature,
and adjust the Rayleigh number 共based on the horizontal
scale兲 to allow the nonlinear inertial terms to be neglected to
first order. With those goals a reference scale of velocity is
suggested by the balance between the viscous force associated with a horizontal scale L and the buoyancy,
共4兲
Here T ref is a reference temperature which will be fixed later.
With q⫽(u,w), we define dimensionless variables,
共 x * ,y * 兲 ⫽ 共 x/L,z/H 兲 ,
w * ⫽w/U ref ,
t * ⫽tL/U ref ,
T * ⫽T/T ref ,
u * ⫽uH/ 共 LU ref兲 ,
共5兲
p * ⫽pH/ 共 ␳ U ref␯ 兲 .
Dropping stars, the dimensionless equations become
␦ 共 u t ⫹uu x ⫹wu z 兲 ⫹p x ⫺u xx ⫺␭ 2 u zz ⫽0,
共6兲
␦ 共 w t ⫹uw x ⫹ww z 兲 ⫹␭ 2 p z ⫺w xx ⫺␭ 2 w zz ⫽T,
共7兲
T t ⫹uT x ⫹wT z ⫽ ␦ ⫺1 P r⫺1 共 T xx ⫹␭ 2 T zz 兲 ,
共8兲
u x ⫹w z ⫽0.
共9兲
Here ␦ is a Reynolds number based upon ␭U ref , L, and ␯ ,
P r is the Prandtl number based upon ␯ , and ␭ is a slenderness parameter:
␦ ⫽g ␣ L 4 T ref / 共 H ␯ 兲 ⫽U refL␭/ ␯ ,
␭⫽H/L,
P r⫽ ␯ / ␬ T .
R a ⫽g ␣ H 3 T ref / 共 ␯ ␬ T 兲 ,
共12兲
where ␬ T is the 共constant兲 thermal diffusivity.
We shall assume that
␦ Ⰶ1,
␭Ⰶ1,
␦ P r Ⰷ1.
共13兲
The last condition is essential, for we shall be neglecting the
diffusion of temperature throughout.
A third small parameter, the amplitude ⑀ of the perturbation of the basic state, is introduced below. The relative
ordering of the assumed small parameters is not immediately
obvious and will be discussed below. We use ⑀ as the basic
ordering parameter.
B. The unperturbed plumes
1
qt ⫹q"“q⫹ “ p⫺ ␯ ⵜ 2 q⫽g ␣ Tiz ,
␳
U ref⫽g ␣ T refL 2 ␯ .
␦ ⫽␭ 4 R a / P r ,
共10兲
共11兲
The parameter ␦ may also be expressed, in terms of the
Rayleigh number R a , as
We define the reduced system to be 共6兲–共9兲 with ␭
⫽ ␦ ⫺1 P r⫺1 ⫽0. This simpler system, which we be used to
discuss perturbations of slender nonconductive temperature
fields, is then
␦ 共 u t ⫹uu x ⫹wu z 兲 ⫹ p x ⫺u xx ⫽0,
␦ 共 w t ⫹uw x ⫹ww z 兲 ⫺w xx ⫽T,
T t ⫹uT x ⫹wT z ⫽0,
共14兲
u x ⫹w z ⫽0.
We shall say ⌺⫽(u,w,p,T) is admissible provided that 共a兲
⌺ solves 共14兲, 共b兲 ⌺ is periodic with period 2 in x, and 共c兲 w
has zero net vertical mass flux,
具w典⬅
冕
2
共15兲
w dx⫽0.
0
An unperturbed plume will be an admissible ⌺⫽⌺ 0 of the
form „u 0 ,w 0 ,p 0 ,T 0 )⫽(0,w 0 (x),Gz,T 0 (x)…. Thus T 0 (x)
may be chosen to be any periodic function with period 2 and
G⫺d 2 w 0 /dx 2 ⫽T 0 共 x 兲 ,
冕
w 0 共 x⫹2 兲 ⫽w 0 共 x 兲 ,
共16兲
2
0
w 0 dx⫽0.
The constant pressure gradient G is needed to balance any
net buoyancy of the assumed temperature profile, and thus
allow condition 共15兲 to be satisfied. In fact, we see immediately upon integration over an interval of periodicity that
G⫽
1
2
冕
2
0
T 0 dx.
共17兲
We remark that, in particular cases discussed below, we will
impose additional restrictions on the positions of the critical
points where w 0 vanishes. This is because perturbations can
change the plume streamline topology near such points.
A simple example of a T 0 is the ‘‘top-hat’’ temperature
profile:
T 0共 x 兲 ⫽
再
1,
if 0⭐ 兩 x 兩 ⬍ ␮ ,
0,
if ␮ ⭐ 兩 x 兩 ⬍1.
共18兲
Here ␮ ,0⬍ ␮ ⬍1, is the half-width of the upward plume of
temperature T ref , as a fraction of the period. Here G⫽ ␮ and
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Chaos, Vol. 10, No. 1, 2000
再
⫺d 2 w 0 /dx 2 ⫽
Instability of convective plumes
共 1⫺ ␮ 兲 ,
⫺␮,
if 0⭐ 兩 x 兩 ⬍ ␮ ,
if ␮ ⭐ 兩 x 兩 ⬍1.
共19兲
Solving with the conditions 共16兲, the unique continuously
differentiable solution is
w 0⫽
再
共 ␮ ⫺1 兲 x 2 /2⫹C,
if 0⭐ 兩 x 兩 ⬍ ␮ ,
␮ 共 x⫺1 兲 /2⫹ ␮ 共 ␮ ⫺1 兲 /2⫹C,
if ␮ ⭐ 兩 x 兩 ⬍1,
2
共20兲
where
C⫽ ␮ 共 ␮ ⫺1 兲共 ␮ ⫺2 兲 /6.
共21兲
We note that here w 0 ( ␮ )⬅w ␮ ⫽ ␮ ( ␮ ⫺1)(2 ␮ ⫺1)/3. If ␮
⬍1/2, w 0 vanishes at 1⫺ 冑(1⫺ ␮ 2 )/3⬎ ␮ , while if ␮
⬎1/2, w 0 vanishes at 冑␮ (2⫺ ␮ )/3⬍ ␮ .
The top-hat profile is very useful for explicit computations, as we shall see below. Note that 共18兲 is a case of
broken TB symmetry, except for the special case ␮ ⫽1/2.
That case is special in another way, for then the discontinuity
of temperature occurs at the zero of w 0 .
A more general plume configuration is the ‘‘double-hat’’
profile,
T 0⫽
再
1,
if 0⭐ 兩 x 兩 ⬍ ␮ ,
0,
if ␮ ⭐ 兩 x 兩 ⬍1⫺ ␯ ,
⫺␣,
共22兲
if 1⫺ ␯ ⭐ 兩 x 兩 ⬍1,
where 0⬍ ␮ ⬍ ␯ ⬍1. Here ␣ is a second parameter. It is convenient to denote the top-hat profile by T (1)
0 (x; ␮ ) and then
write 共22兲 as
(1)
(1)
T (2)
0 ⫽T 0 共 x; ␮ 兲 ⫺ ␣ T 0 共 x⫺1; ␯ 兲 .
共23兲
(1)
(1)
w (2)
0 ⫽w 0 共 x; ␮ 兲 ⫺ ␣ w 0 共 x⫺1; ␯ 兲 ,
共24兲
The explicit form is
w 0⫽
再
共 ␮ ⫺ ␣ ␯ ⫺1 兲 x 2 /2⫹A,
if 0⭐ 兩 x 兩 ⬍ ␮ ,
共 ␮ ⫺ ␣ ␯ 兲 x /2⫹Bx⫹C,
if ␮ ⭐ 兩 x 兩 ⬍1⫺ ␯ ,
2
共 ␣ ⫹ ␮ ⫺ ␣ ␯ 兲共 x⫺1 兲 2 /2⫹D,
共25兲
if 1⫺ ␯ ⭐ 兩 x 兩 ⬍1,
where
A⫽ ␮ 3 /6⫺ ␣ ␯ 3 /6⫺ ␮ /6⫹ ␣ ␯ /6⫹ ␮ 共 1⫺ ␮ 兲 /2,B⫽⫺ ␮ ,
共26兲
C⫽A⫹ ␮ 2 /2, D⫽A⫹ ␮ 共 ␮ ⫺1 兲 /2⫹ ␣ ␯ 共 ␯ ⫺1 兲 /2.
The case of TB symmetry is ␣ ⫽1,␯ ⫽ ␮ :
w 0⫽
are of interest. The first will be termed a ‘‘shift’’ instability,
since it will result in a horizontal displacement of all fluid
particles, equally at the same value of z. A new steady state
obtained from a shift instability would have streamlines
which are equivalent under parallel displacement in x, and
cannot involve any Eulerian mean flow. However, we shall
then identify a second instability which is associated with an
Eulerian mean flow. In a new steady state obtained from the
second instability, streamlines of upward moving and downward moving particles have opposite deflections. This can be
understood by considering a simple combination of a plume
„0,w 0 (x)… and a small mean flow ⑀ sin 2␲z. The perturbed
streamline passing through (x 0 ,0) is given by
x⫽ ⑀ 关 2 ␲ w 0 l 共 x 0 兲兴 ⫺1 sin 2 ␲ z,
再
⫺x 2 /2⫹ ␮ 共 1⫺ ␮ 兲 /2,
⫺ ␮ 共 x⫺1/2兲 ,
if 0⭐ 兩 x 兩 ⬍ ␮ ,
if ␮ ⭐ 兩 x 兩 ⬍1⫺ ␮ ,
⫹ 共 x⫺1 兲 /2⫺ ␮ 共 1⫺ ␮ 兲 /2,
2
共27兲
if 1⫺ ␮ ⭐ 兩 x 兩 ⬍1.
We point out that the double-hat profile provide a model of a
convective field combining what can be described as distinct
up- and down-plumes.
C. Remarks concerning instabilities of plumes
In the subsequent analysis we study instabilities of a
slender plume. We shall show that two distinct instabilities
共28兲
provided that w 0 (x 0 )⫽0. Note that, in the vicinity of any
zero of a zero of w 0 (x), ‘‘cat’s-eye’’ regions of closed
streamlines are formed 共see Fig. 2 below兲. Our object is to
discuss both kinds of instability and see how they may be
combined to understand the emergence of a mean flow from
a perturbed plume.
III. SHIFTED SOLUTIONS
A. Shift invariance
A key ingredient in our analysis, for admissible solutions
of 共14兲, is a property of the plumes derived from the slenderness assumption. It is analogous to a well-known ‘‘shift
invariance’’ of Prandtl’s two-dimensional boundary-layer
equations. These latter equations are
u t ⫹uu x ⫹ v u y ⫹ p x ⫺u y y ⫽0,
Thus
31
p⫽ p 共 x,t 兲 ,
u x ⫹ v y ⫽0.
共29兲
It is easily seen that, if „u(x,y,t), v (x,y,t)… solves 共29兲 then
so does
u s ⫽u 共 x,y s ,t 兲 ,
p s 共 x,t 兲 ⫽ p,
v s ⫽ v共 x,y s ,t 兲 ⫹ ␩ t ⫹u 共 x,y s ,t 兲 ␩ x ,
共30兲
where y s (x,y,t)⫽y⫺ ␩ (x,t), ␩ being an arbitrary function.
The proof uses the chain rules,
⳵ t 兩 x,y ⫽ ⳵ t 兩 x,y s ⫺ ␩ t ⳵ y s 兩 x,t ,
⳵ x 兩 t,y ⫽ ⳵ x 兩 t,y s ⫺ ␩ x ⳵ y s 兩 x,t .
共31兲
This can also be expressed by a transformation on the stream
function, since (u s , v s )⫽( ⳵ y ␺ s ,⫺ ⳵ x ␺ s ) where ␺ s ⫽ ␺ (x,y
⫺ ␩ ,t)⫺ 兰 ␩ t dx.
We then have
u sx ⫹ v sy ⫽⫺u y 共 x,y⫺ ␩ ,t 兲 ␩ x ⫹u y 共 x,y⫺ ␩ ,t 兲 ␩ x ⫹C⫽0,
共32兲
u st ⫹u s u sx ⫹ v s u sy ⫹ p sx ⫺u sy y
⫽⫺u y 共 x,y s ,t 兲共 ␩ t ⫹u 共 x,y s ,t 兲 ␩ x 兲
⫹u y 共 x,y s ,t 兲共 ␩ t ⫹u 共 x,y s ,t 兲 ␩ x 兲 ⫹M 兩 y⫽y s ⫽0,
共33兲
where C⫽u x ⫹ v y and M ⫽u t ⫹uu x ⫹ v u y ⫹ p x ⫺u y y , each
evaluated at x,y s ,t, which completes the proof.
The function ␩ (x,t) determines a vertical shift of the
entire flow field . The shifted boundary layer may be thought
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32
Chaos, Vol. 10, No. 1, 2000
Stephen Childress
of as essentially parallel at any instant to a curve y
⫽ ␩ (x,t), away from its base position y⫽0. This shift invariance expresses the thinness of the boundary layer in
physical variables, as well as the fact that the only viscous
stress in the problem derives from the cross-stream variation
of the velocity. This property figures prominently in the
analysis of boundary-layer separation; see Ref. 14.
We shall state a corresponding result for 共14兲. Since our
plumes are vertical the shift will now be in the horizontal
coordinate x.
Theorem: Let ⌺⫽(u,w, p,T) be any admissible solution of the reduced equations (14). Then ⌺ s
⫽(u s ,w s ,p s ,T s ) is also a solution, where
u s ⫽u 共 x⫺ ␰ ,z,t 兲 ⫹ ␰ t ⫹w 共 x⫺ ␰ ,z,t 兲 ␰ z ,
共34兲
w s ⫽w 共 x⫺ ␰ ,z,t 兲 ,
p s⫽ ⳵ xu s⫹ ␦
冕
G s dx,
T s ⫽T 共 x⫺ ␰ ,z,t 兲 ,
共35兲
共36兲
provided that ␰ (z,t) satisfies
␰ tt ⫹„␹ 共 z,t 兲 ␰ z …z ⫽0,
␹ 共 z,t 兲 ⫽ 具 w 2 典 ⬅
冕
2
0
共37兲
w 2 共 x,z,t 兲 dx.
This result states an invariance property under the condition
共37兲 on ␰ , and therefore goes further than Prandtl boundary
layer shift by including a constraint imposed by the
x-momentum balance.
To prove the theorem we again may verify that the
shifted solution ⌺ s is a solution of the system, by substitution
using formulas analogous to 共31兲. The last three equations of
共14兲, expressed in the new variables, may be shown to hold
as in the shift of the Prandtl boundary layer. The new feature
is the calculation of p s , which enforces a condition that G s
have zero horizontal mean. We then have
具 G s 典 ⫽ 具 u 典 t ⫹ 具 uw 典 z ⫹ ␰ tt ⫹ 具 w ␰ z 典 t ⫹ 具 ␰ t w 典 z ⫹ 共 具 w 2 典 ␰ z 兲 z ⫽0.
共38兲
Since ⌺ solves 共14兲, the average of the reduced x-momentum
equation yields 具 u 典 t ⫹ 具 uw 典 z ⫽0. Since w satisfies 共15兲,
具 w ␰ z 典 t ⫽ 具 ␰ t w 典 z ⫽0. These expressions imply 共37兲 and establish the theorem.
B. Shift instability
Applying Theorem 1 to an unperturbed plume ⌺ 0 yields
the following result: To linear terms in ␰ , 共37兲 yields
␰ tt ⫹ 具 w 20 典 ␰ zz ⫽0,
␰ tt ⫹ 具 w 20 典 ␰ zz ⫺ ␥ ␰ zzt ⫽0,
共39兲
since w 0 is independent of z. Thus, with ␰ ⫽e ikz⫹ ␴ t we extract an instability having growth rate ␴ s ⫽ 兩 k 兩 冑具 w 20 典 . The
growth of ␴ with k indicates a poorly posed initial-value
problem, which will have to be resolved outside the reduced
equations. We note in passing one way which this can be
done. Suppose that we take 共6兲 in its entirety for the
x-momentum equation of the ‘‘reduced system,’’ while still
␥ ⫽␭ 2 / ␦ .
共40兲
In this case the shift instability has the growth rate
␴ s 共 k 兲 ⫽ 21 关 ⫺k 2 ␥ ⫹ 冑k 4 ␥ 2 ⫹4 具 w 20 典 兴 ,
共41兲
具 w 20 典 / ␥
which grows monotonically from 0 to
as 兩 k 兩 increases from 0, thus making the initial-value problem well
posed.
Since we shall be investigating the linear instability of
an unperturbed plume, we note that, if ⌺⬇⌺ 0 ⫹⌺ ⬘ is the
linear approximation in powers of ␰ , we have
冉
⌺ ⬘ ⫽ ␰ t ⫹w 0 ␰ x ,⫺w 0x ␰ ,w 0x ␰ z
⫹
where ␰ is a function of z,t and
G s ⫽ ⳵ t u s ⫹u s ⳵ x u s ⫹w s ⳵ z u s ,
omitting all other terms in ␭ 2 . This is of course not a fully
consistent procedure but it slightly enlarges the reduced system. Then Theorem 1 again obtains with an additional term
in G s and an altered equation for ␰ :
冕
冊
共 2w 0 ␰ zt⫹w 20 ⫺w̄ 20 兲 dx,⫺T 0x ␰ ,
共42兲
given that ␰ satisfies 共37兲.
Since our aim in this paper is an understanding of the
origin of mean flow in this model, it is important to observe
that the mean flow in the shift instability, ␰ t , is a transient
which disappears if and when ␰ becomes stationary.
IV. LINEAR ANALYSIS AND THE MEAN-FLOW
INSTABILITY
In this section we shall discuss a bifurcation from the
unperturbed plumes which is not a shift instability. We examine this by linearizing 共6兲–共9兲. Let the expansion in ⑀ for
fixed ␦ ,␭ be ⌺⫽(u,w,p,T)⫽⌺ 0 ⫹ ⑀ ⌺ ⬘ ⫹ . . . , ⑀ Ⰶ1. The
linearized equations are then, dropping primes,
␦ 共 u t ⫹w 0 u z 兲 ⫹ p x ⫺u xx ⫺␭ 2 u zz ⫽0,
␦ 共 w t ⫹w 0 w z ⫹u dw 0 /dx 兲 ⫹␭ 2 p z ⫺w xx ⫺␭ 2 w zz ⫽T,
T t ⫹w 0 T z ⫹u dT 0 /dx⫽0,
共43兲
u x ⫹w z ⫽0.
If we set ␦ ⫽␭⫽0 in 共43兲 we have the limiting system
p x ⫺u xx ⫽0,
⫺w xx ⫽T,
w 0 T z ⫹u dT 0 /dx⫽0,
共44兲
u x ⫹w z ⫽0.
We shall now consider in some detail the solutions of 共44兲.
Since, from 共44兲, we have ⫺w 0 w xxz ⫽w 0 u xxx ⫽w 0 T z
⫽⫺T t ⫺u dT 0 /dx and ⫺d 2 w 0 /dx 2 ⫽T 0 , u,w satisfy
w 0 u xxx ⫽w xxt ⫹uw 0xxx ,
u x ⫹w z ⫽0.
共45兲
Thus we see that the linearized shift solution u⫽ ␰ t (z,t)
⫹ ␰ z (z,t)w 0 (x), w⫽⫺ ␰ (z,t)w 0x , T⫽⫺ ␰ (z,t)T 0x is one
solution of 共45兲. We now seek other solutions, under the
additional assumption that they do not depend upon t, so that
共45兲 provides a single equation for u,
w 0 u xxx ⫽uw 0xxx .
共46兲
In other words, we now seek linear instability modes on a
time scale long compared to the natural time scale of the
shift instability, the latter given by the solution u⫽⫺ ␰ z w 0 .
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Chaos, Vol. 10, No. 1, 2000
Instability of convective plumes
33
The question we ask is, are there any other solutions of 共46兲?
In order to eliminate additive solutions of the form
(u,w,p,T)⫽„0,f (x),0,⫺ f xx … we shall in what follows
specify the perturbation to have a mean in z which is zero.
A. Specific cases
The answer to the last question depends upon the choice
of w 0 , so we first examine some cases explicitly. Consider
first the top-hat profile 共18兲. Here 共46兲 takes the form
w 0 u xxx ⫽u 关 ␦ 共 x⫺ ␮ 兲 ⫺ ␦ 共 x⫹ ␮ 兲兴 ,
⫺1⭐x⬍⫹1. 共47兲
It is easily seen by direct calculation that any continuously
differentiable solution of 共47兲 is a multiple of w 0 and thus a
shift solution. Indeed we must have u xxx ⫽0, x⫽⫾ ␮ , together with the jump conditions
关 u xx 兴共 ␮ 兲 ⫽u 共 ␮ 兲 /w ␮ ,
关 u xx 兴共 ⫺ ␮ 兲 ⫽⫺u 共 ⫺ ␮ 兲 /w ␮ ,
共48兲
关 • 兴 denoting the jump from left to right and w ␮ ⫽w 0 ( ␮ )
⫽ ␮ ( ␮ ⫺1)(2 ␮ ⫺1)/3. Solving with
u⫽
再
A 共 x⫹1 兲 ⫹B 共 x⫹1 兲 ⫹C,
2
Dx ⫹Ex⫹F,
2
if ⫺1⭐x⬍⫺ ␮ ,
if ⫺ ␮ ⭐x⬍ ␮ ,
A 共 x⫺1 兲 ⫹B 共 x⫺1 兲 ⫹C,
2
共50兲
F⫽ 共 ␮ ⫺1 兲共 ␮ ⫺2 兲 A/3.
We thus find that 共49兲 is 2w 0 / ␮ , so the only steady solution
is a shift.
To see that this is not always true, we consider next the
double-hat profile with ␣ ⫽1, ␯ ⫽ ␮ . The equation for u is
then
3U
u⫽
2 共 2 ␮ ⫹1 兲
再
再
if 0⭐x⬍ ␮ ,
Ax 2 ⫹B,
if ␮ ⭐x⬍1⫺ ␮ ,
C 共 x⫺1/2兲 ⫹D,
2
共52兲
if 1⫺ ␮ ⭐x⬍1,
A 共 x⫺1 兲 ⫹B,
2
and extend this as an even function about 0, u(⫺x)
⫽u(x),)⭐x⬍1. The conditions are that u,u x be continuous
at x⫽ ␮ , 1⫺ ␮ , that 关 u xx 兴 ( ␮ )⫽u( ␮ )/w ␮ , 关 u xx 兴 (1⫺ ␮ )
⫽⫺u(1⫺ ␮ )/w ␮ , and finally that
冕
1
共53兲
u dx⫽U,
0
a given number.
A unique solution of the last problem, having the form
共52兲, can be found:
if 0⭐x⬍ ␮ ,
⫺2 共 x⫺1/2兲 / 共 2 ␮ ⫺1 兲 ⫹ ␮ ⫹1/2,
2
⫺ 共 x⫺1 兲 / ␮ ⫹ ␮ ⫹1,
2
共51兲
Here w ␮ ⫽ ␮ (1⫺2 ␮ )/2. Since w 0 is now odd in x about x
⫽1/2, we look for a solution u which is even with respect to
x⫽1/2. Assume
u⫽
D⫽ 共 1⫺1/␮ 兲 A,
⫺x 2 / ␮ ⫹ ␮ ⫹1,
u xxx ⫺uw ␮⫺1 关 ␦ 共 x⫺ ␮ 兲 ⫺ ␦ 共 x⫹1⫺ ␮ 兲兴 ⫽0.
共49兲
if ␮ ⭐x⬍1,
where we have already imposed the periodicity conditions
u(⫺1)⫽u(⫹1), u x (⫺1)⫽u x (⫹1), u xx (⫺1)⫽u xx (⫹1),
we obtain B⫽E⫽0,
C⫽ 共 ␮ 2 ⫺1 兲 A/3,
FIG. 1. The normalized eigenfunctions of the double hat for ␮⫽0.3.
if ␮ ⭐x⬍1⫺ ␮ ,
共54兲
if 1⫺ ␮ ⭐x⬍1.
Let us write the equation 共51兲 as Lu⫽0, and adopt the L 2
inner product on the interval 共⫺1,1兲, with norm 储•储. Setting
u e1 ⫽w 0 / 储 w 0 储 and u e2 ⫽u/ 储 u 储 , where u is given by 共54兲, the
eigenspace of the zero eigenvalue of L for the symmetric
double hat is thus spanned by u e1 ,u e2 . Both eigenfunctions
are even in x.
The adjoint operator to L, L * is defined by
L * u⫽⫺u xxx ⫺uw ␮⫺1 关 ␦ 共 x⫺ ␮ 兲 ⫺ ␦ 共 x⫹1⫺ ␮ 兲兴 ⫽0.
共55兲
Since the geometric multiplicity of the eigenvalue 0 is the
same for L and L * , for the symmetric double hat we can find
* ,u e2
* given by
two distinct eigenfunctions u e1
*⫽
u⫽u e1
*⫽
u⫽u e2
再
冦
共 2 ␮ ⫺1 兲 x,
if 0⭐x⬍ ␮ ,
共 x⫺1/2兲 ⫹ ␮ 2 ⫺1/4,
2
⫺ 共 2 ␮ ⫺1 兲共 x⫺1 兲 ,
⫺x 2 / ␮ ⫹ ␮ ⫹1,
if ␮ ⭐x⬍1⫺ ␮ ,
if 1⫺ ␮ ⭐x⬍1;
共56兲
if 0⭐x⬍ ␮ ,
⫺2 共 x⫺1/2兲 / 共 2 ␮ ⫺1 兲 ⫹ ␮ ⫹1/2,
2
if ␮ ⭐x⬍1⫺ ␮ ,
⫺ 共 x⫺1 兲 2 / ␮ ⫹ ␮ ⫹1,
if 1⫺ ␮ ⭐x⬍1.
共57兲
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34
Chaos, Vol. 10, No. 1, 2000
* is odd in x and u e2
* is even in x, and that only
Note that u e1
* have nonzero means. We show normalized 共in the
u e2 ,u e2
above norm兲 eigenfunctions in Fig. 1. In Fig. 2 we show the
streamlines of the perturbed symmetric double-hat plume
Q 0 ⫹ ⑀ Q e2 with ⑀⫽0.05 and U⫽⫺sin ␲z.
The geometric multiplicity of the eigenvalue 0 is in fact
equal to two for a family of double-hat plumes. These were
obtained numerically using MATLAB routines. If we fix ␮
⫽0.3 and vary ␯ ,a, the profiles in Table I have multiplicity
two.
The symmetric double-hat is special in that u e2 has a
first derivative which vanishes at x⫽⫾1/2, and this is a
condition not satisfied by u e1 in general. If we add the con⫹1/2
u dx⫽0, we obtain, as is easily checked, a probdition 兰 ⫺1/2
lem for perturbations of the symmetric double hat which
allows only the trivial solution u⫽0. Thus we can regard the
eigensolution as ‘‘forced’’ by the assumed nonvanishing
mean value. We then treat the linear problem in the reduced
interval 兩 x 兩 ⬍1/2 with perturbation u 1 ⫽U⫹u 1 , where
⳵ u 1 / ⳵ x(⫾1/2)⫽0, 兰 10 u 1 dx⫽0; then u 1 is found by a forced
linear problem. This opens the way to a much simpler analysis than would be realized by bifurcation theory at a degenerate eigenvalue. We shall term the reduced interval the
‘‘cell’’ of the computation, and the resulting perturbation a
theory of the ‘‘plume-in-cell.’’ In effect we have isolated a
problem where a single upwelling plume may be subjected to
an arbitrary mean flow U imposed on the leading perturbation u 1 , with zero mean flow imposed on all higher-order
terms in the expansion of u. In this way the shift instability is
expelled from the problem. Moreover, as long as the higherorder theory focuses on the reduced equations 共14兲, the shift
instability can be added at any time.
B. Linear theory of plume-in-cell: Subscript 2
We now turn to the expansion of the mean-flow mode
we have obtained above with respect to ␦ ,␭⬃ ⑀ , assuming
the flow is steady. If we expand ⌺ ⬘ satisfying 共43兲 as ⌺ ⬘
⫽⌺ 1 ⫹ ␦ ⌺ 2 ⫹O( ⑀ 2 ) we obtain
Stephen Childress
⫺d 2 w 2 /dx 2 ⫺T 2 ⫽⫺w 0 ⳵ w 1 / ⳵ z⫺u 1 dw 0 /dx,
共58兲
w 0 ⳵ T 2 / ⳵ z⫹u 2 dT 0 /dx⫽0.
Thus
Lu 2 ⫽w 0 ⳵ 2 u 1 / ⳵ x ⳵ z⫺ 共 ⳵ u 1 / ⳵ z 兲 dw 0 /dx⬅ f 2 .
共59兲
Working now in the interval 兩 x 兩 ⬍1/2, we have 关cf. 共54兲兴
3U
u⫽u 1 ⫽
2 共 2 ␮ ⫹1 兲
With
w 0⫽
再
再
⫺x 2 / ␮ ⫹ ␮ ⫹1,
⫺2 共 x⫺1/2兲 / 共 2 ␮ ⫺1 兲 ⫹ ␮ ⫹1/2,
if ␮ ⭐ 兩 x 兩 ⬍1/2.
⫺x 2 /2⫹ ␮ 共 1⫺ ␮ 兲 /2,
⫺ ␮ 共 x⫺1/2兲 ,
if 0⭐ 兩 x 兩 ⬍ ␮ ,
2
共60兲
if 0⭐ 兩 x 兩 ⬍ ␮ ,
if ␮ ⭐ 兩 x 兩 ⬍1/2,
共61兲
we see that
f 2⫽
再
2␮x
if 0⭐ 兩 x 兩 ⬍ ␮ ,
2␮
共 x 2 ⫺x⫹ ␮ 2 兲
2 ␮ ⫺1
if ␮ ⭐ 兩 x 兩 ⬍1/2.
共62兲
Solving 共59兲 with 共62兲 subject to the conditions ⳵ u 2 / ⳵ x
(⫾1/2)⫽0, 兰 10 u 2 dx⫽0, we obtain u 2 and can compute the
leading approximation to 具 uw 典 :
具 uw 典 ⬃ ⑀ ␦ 具 u 2 w 0 典 ⬅ ⑀ ␦ ␬ 1 共 ␮ 兲 DU.
共63兲
We show the function ␬ 1 / 储 w 0 储 2 in Fig. 3. Note that values
out to ␮⬇0.38 are positive, which will enable an instability
in the mean-flow equation. The normalization using w 0 in
this figure is dictated by the rather small values of w 0 obtained from the equilibrium solution, i.e., the fact that U ref is
not a very good indication of the actual size of the unperturbed vertical velocity.
V. THE EQUATION FOR THE MEAN FLOW
We now turn to the derivation of the full expansion of
the perturbed plume-in-cell, that is, for the symmetric double
hat with zero shift, and to the calculation of the form of the
nonlinear equation governing the evolution of the mean flow.
We accordingly restrict attention to plumes with this symmetry and to the interval 兩 x 兩 ⭐1/2.
The equations are then
␦ 共 u t ⫹uu x ⫹wu z 兲 ⫹ p x ⫺u xx ⫺␭ 2 u zz ⫽0,
共64兲
␦ 共 w t ⫹uw x ⫹ww z 兲 ⫹␭ 2 p z ⫺w xx ⫺␭ 2 w zz ⫽T,
共65兲
T t ⫹uT x ⫹wT z ⫽ ␦ ⫺1 P r⫺1 共 T xx ⫹␭ 2 T zz 兲 ,
u x ⫹w z ⫽0.
共66兲
TABLE I. Values of ␯ and a where the null space of the eigenvalue 0 is
two-dimensional.
FIG. 2. Streamlines of Q⫹0⫹ ⑀ Q e2 where Q 0 is the symmetric double-hat
profile ␮ ⫽ ␯ ⫽0.3, a⫽1, ⑀ ⫽0.05, and the mean flow is U⫽⫺sin ␲z.
a
0.5
0.7
1
1.3
1.5
2
␯
0.41764
0.36570
0.30000
0.24984
0.22355
0.17542
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Chaos, Vol. 10, No. 1, 2000
Instability of convective plumes
35
TABLE II. Properties of contributing terms to the fifth-order mean-flow
equation. Here Û denotes the indefinite integral of U, and D⫽ ⳵ / ⳵ z. Also
A m ⫽w 0 u mxz ⫺u mz w 0x , B mn ⫽D(u m w nx ⫹w m Dw n ), C mn ⫽w ⫺1
0 (u m T nx
⫹w m DT n ). A ‘‘*’’ in the first column indicates that the contribution vanishes under integration over full period 2 by top–bottom symmetry. The
‘‘†’’ indicates that u n is odd with respect to x⫽0, and so gives no contribution to 具 w 0 u n 典 . p n is obtained from the expansion of the x-momentum
equation and is not given explicitly here.
Subscript
n
FIG. 3. The parameter ␬ 1 / 储 w 储 2 as a function of ␮.
As a working hypothesis we shall take ⑀ , ␦ ,␭ to be small and
of the same relative order, measured in powers of ⑀ . We
consider the expansion through terms of formal order ⑀ 4 :
1*
2
3†
4*
5*
6*†
7*
8
9
10†
11†
12
13†
Order
Functional
dependence
of u n on U
fn
⑀
⑀␦
⑀2
⑀␭2
⑀␦2
⑀ 2␦
⑀3
⑀␦␭2
⑀␦3
⑀ 2␭ 2
⑀ 2␦ 2
⑀ 3␦
⑀4
U
DU
D(Û 2 )
D 2U
D 2U
D 2 (U 2 )
DÛ 3
D 3U
D 3U
D 3 (Û 2 )
D 3 (Û 2 )
D 2 (Û 2 )
D(Û 4 )
0
A1
⫺C 11
⫺p 1zz ⫺u 1xzz
A2
A 3 ⫺B 11⫺C 12⫺C 21
⫺C 13⫺C 31
⫺p 2zz ⫺u 2xzz ⫹A 4
A5
⫺p 3zz ⫺u 3xzz ⫺C 14⫺C 41
A 6 ⫺B 12⫺B 21⫺C 22⫺C 15⫺C 51
A 7 ⫺B 13⫺B 31⫺C 23⫺C 32⫺C 16⫺C 61
⫺C 33⫺C 17⫺C 71
⌺⫽⌺ 0 ⫹ ⑀ ⌺ 1 ⫹ ⑀ ␦ ⌺ 2 ⫹ ⑀ 2 ⌺ 3 ⫹ ⑀ ␭ 2 ⌺ 4
⫹ ⑀ ␦ 2⌺ 5⫹ ⑀ 2 ␦ ⌺ 6⫹ ⑀ 3⌺ 7
共67兲
⫹ ⑀ ␦ ␭ 2 ⌺ 8 ⫹ ⑀ ␦ 3 ⌺ 9 ⫹ ⑀ 2 ␭ 2 ⌺ 10
⫹ ⑀ ␦ 2 共 ␬ 8 ␭ 2 ⫹ ␬ 9 ␦ 2 兲 D 4 U⫽O 共 ⑀ 5 兲 ,
⫹ ⑀ ␦ ⌺ 11⫹ ⑀ ␦ ⌺ 12⫹ ⑀ ⌺ 13⫹o 共 ⑀ 兲 .
2
2
3
4
⑀ ␦ U t ⫹N 共 U 兲 ⫹ ␦ ⑀ 共 ␬ 1 ␦ 2 ⫺␭ 2 兲 D 2 U
4
共68兲
The linear theory generates the six terms linear in ⑀ , and
⌺ 1,2 , have been discussed, the former introducing the variable U(z, ␶ ), where we now include an as yet unspecified
slow time variable ␶ .
For each term we may derive a governing equation
Lu n ⫽ f n where L is defined by 共51兲, which is to be solved for
1/2
u n dx⫽0, ⳵ u n / ⳵ x
u n subject to the conditions 兰 ⫺1/2
(⫾1/2,z)⫽0. We are interested here in the z-dependence of
f n as this will determine the form of the contribution of 具 uw 典
to the mean-field equation for U(z, ␶ ). The various forcing
terms and the functional contribution to the mean-field equation is given in Table II, where we now let D⫽ ⳵ z :
The ‘‘*’’ indicating that the term is not present with
top–bottom symmetry can be determined by summing the
numbers of ˆ and D symbols. The result must be odd if the
terms survives with top–bottom symmetry. The coefficient
of such a term is a pseudo-scalar with respect to this symmetry. The ‘‘†’’ identifies canceling symmetry with respect
to x⫽0. For example, f 3 contains terms, such as
⫺u 1 T 1x , which are even with respect to x⫽0, so u 3 must be
odd. In determining the functional dependence on U we have
used the fact that, since u nx ⫹Dw n ⫽0, the z dependence of
u n must be of the form of D of a functional of U.
The mean-field equation for the symmetric double hat,
through fourth-order terms, can thus be obtained from the
following generally nonvanishing contributions to the Reynolds stress: 具 w 0 u n 典 ⫽ ␬ n , n⫽2,8,9,12; 具 w 1 u n 典 ⫽ ␬ (1)
n , n
⫽6; 具 w 2 u n 典 ⫽ ␬ (2)
n , n⫽3. This leads to the following form
for the mean-field equation:
共69兲
where
2 2
N 共 U 兲 ⫽ ⑀ 3 ␦ 2 关 ␬ 12D 3 共 Û 3 兲 ⫹ ␬ (1)
6 D 共 ÛD Û 兲
⫹D 共 UDÛ 2 兲兴 .
共70兲
Now to obtain terms of order ⑀ throughout we must make ␦
and ␭ of the same order and take ␬ 1 ␦ 2 ⫺␭ 2 to be of order ⑀ 2 .
Recall that we have established the positivity of ␬ 1 for sufficiently small ␮ ; see Fig. 3. We also set ⑀ 3 t⫽ ␶ to define the
slow time. Neglecting terms of order ⑀ 6 or higher, the equation then assumes the form
5
U ␶ ⫹D 共 ␩ 1 Û 2 DU⫹ ␩ 2 ÛU 2 ⫹ ␹ 1 DU⫹ ␹ 2 D 3 U 兲 ⫽0,
共71兲
⫺3
where ␩ m , ␹ m are O(1) constants and ␹ 1 ⫽ ␬ 1 ⑀ ( ␦ 2
⫺␭ 2 ).
We have no knowledge of the constants in 共71兲 except
for ␹ 1 . If the instability is to be cut off at a large wavenumber to the order considered here, it is necessary that ␹ 2 ⬎0.
共Otherwise the instability can be cut off only by spatial derivatives of order six or higher.兲 If the equation is multiplied
by U and integrated with respect to x from 0 to 2 we obtain,
after several integrations by parts,
1
d/d ␶
2
冕
2
u 2 dx⫽
0
冕 冋␹
2
0
1 共 DU 兲
2
⫺ ␹ 2共 D 2U 兲 2
册
1
⫹ ␩ 1 Û 2 共 DU 兲 2 ⫺ ␩ 2 U 4 dx.
3
共72兲
Thus if the instability saturates nonlinearly we should have
␩ 1 ⬍0 and ␩ 2 ⬎0.
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36
Chaos, Vol. 10, No. 1, 2000
Stephen Childress
VI. PLUMES IN THREE DIMENSIONS
It is of interest to examine the corresponding problem in
three dimensions, when the plumes are slender cylindrical
objects. The problem is technically more complicated than
the two-dimensional calculations, but the 2D model can nevertheless serve as a useful guide. The complications come
both from the representation of unperturbed plumes and the
form of the homogeneous problem for perturbations within a
plume-in-cell model. We also need to decide how to derive
the latter simplification in three dimensions.
Consider a convecting layer with horizontal coordinates
x,y, “ H denoting the gradient in the horizontal. Let T 0 (x,y)
be some unperturbed temperature field, arbitrary but having a
horizontal average. For computations we might take T 0 as a
linear combination of Fourier modes T k⫽e ik"r,r⫽(x,y),k
⫽(k x ,k y ) over a set of k which includes (0,0). Let w 0 (x,y)
be the corresponding unperturbed plume structure, satisfying
2
w 0 ⫽T 0 , and having zero horizontal aver关cf. 共16兲兴 G⫺ⵜ H
age. The reduced system 关cf. 共14兲兴 is then, writing the velocity as (u, v ,w)⫽(u,w),
2
u⫽0,
e ␦ 共 ut ⫹u"“H u⫹wuz 兲 ⫹“ H p⫺ⵜ H
␦ 共 w t ⫹u"“H w⫹ww z 兲 ⫺ⵜ H2 w⫽T,
T t ⫹u"“H T⫹wT z ⫽0,
共73兲
“ H •u⫹w z ⫽0.
To discuss the shift instability in three dimensions, we confine attention to ␦ ⫽0 and the linearized steady system,
2
u⫽0,
“ H p⫺ⵜ H
2
⫺ⵜ H
w⫽T,u"“H T 0 ⫹w 0 T z ⫽0,
“ H •u⫹w z ⫽0.
共74兲
A shift perturbation solving 共74兲 will have the velocity
共 u, v ,w 兲 ⫽ 共 ␰ z w 0 ⫹ ␺ y , ␩ z w 0 ⫺ ␺ x ,⫺ ␰ w 0x ⫺ ␩ w 0y 兲 , 共75兲
where ␳⫽„␰ (z), ␩ (z)… is the horizontal shift function. The
new streamfunction ␺ is needed to ensure that horizontal
viscous forces are balanced by a pressure. The z-component
of the curl of the horizontal momentum equation yields
4
␺ ⫹ 共 ␰ z ⳵ y ⫺ ␩ z ⳵ x 兲 ⵜ H2 w 0 ⫽0.
ⵜH
共76兲
We may assume that this equation can be solved, given T 0 as
above, for a unique bounded ␺ , and the pressure then has the
form p⫽ ␰ w 0x ⫹ ␩ w 0y ⫹constant.
What is of particular interest is the choice of plume-incell model in three dimensions, allowing a direct analysis of
a mean-flow instability. The main difficulty here is of course
the fact that 2D periodic structures such as square or hexagonal cells are somewhat awkward to treat insofar as stability is
concerned. It is clear that the obvious approximate model is
a cylindrical cell, as in Fig. 4, the domain being 0⬍r⫽ 兩 r兩
⬍1 and, in the case of a top-hat temperature, T 0 ⫽1 in 0
⬍r⬍ ␮ and zero otherwise. If the exact field is replaced by
an array of up and down plumes of this type, an approximate
averaging procedure must be devised to incorporate the results of the plume-in-cell calculation. This averaging can either can be based either on a statistical model, so the parameters of the cell are regarded as conditional expectations
given that attention is restricted to a single plume, or else on
averaging over many cells immersed in a common vertical
FIG. 4. A 3D ‘‘up’’ plume with a top-hat profile, perturbed by a mean flow.
mean pressure gradient. We next give an example of a
plume-in-cell cylindrical geometry of the latter type and suggest that the boundary conditions are consistent in that computations over a range of ␮ indicate a unique solution of the
perturbed fields in the presence of a prescribed horizontal
mean U⫽具u典.
Motivated by the symmetric double hat of the 2D theory,
we imagine an array of equal numbers of equivalent ‘‘up’’
共temperature⫹1兲 and ‘‘down’’ 共temperature⫺1兲 top-hat
plumes in a zero vertical mean pressure gradient. In an ‘‘up’’
plume, for example, we assume w 0 vanishes at r⫽1. For the
top-hat profile, we then have
w 0⫽
再
共 ␮ 2 ⫺r 2 兲 /4⫺ ␮ 2 ln共 ␮ 兲 /2,
⫺ ␮ ln共 r 兲 /2,
2
if ␮ ⭐r⬍1.
if 0⭐r⬍ ␮ ,
共77兲
In the present case the ⑀ -expansion with terms ⌺ n
⫽(un ,w n ,p n ,T n ) produces a system of the form
2
un ⫽fn ,
“ H p n ⫺ⵜ H
共78兲
2
w n ⫺T n ⫽g n ,
⫺ⵜ H
共79兲
un •“ H T 0 ⫹w 0 ⳵ z T n ⫽h n ,
共80兲
“ H •un ⫹ ⳵ z w n ⫽0.
共81兲
Here the forcing terms fn ,g n ,h n are known at each stage,
with un ⫽“ H ␾ n ⫹“ H ⫻iz ␺ n . Note that un ,n⭓2 must in this
case have zero average over the cell, but the leading term
satisfies u v ⫽U to introduce the mean horizontal velocity
field. Also ⌺ is subject to other boundary conditions discussed below.
We now argue that the corresponding homogeneous system,
2
u⫽0,
“ H p⫺ⵜ H
2
ⵜH
w⫹T⫽0,
“ H •u⫹Dw⫽0,
u•“ H T 0 ⫹w 0 DT⫽0,
共82兲
has only the trivial solution for certain temperature profiles
of physical interest. In this case we can obtain a unique expansion similar to the 2D case, with the added feature of a
dependence upon the polar angle ␪. We then have for the
leading term of the horizontal velocity,
u1 ⫽U共 z,t 兲 ⫹“ H ␾ 1 ⫹“ H ⫻iz ␺ 1 .
共83兲
Also, in 共78兲–共81兲, f1 ⫽g 1 ⫽0 but
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Chaos, Vol. 10, No. 1, 2000
Instability of convective plumes
37
The six conditions determine a matrix A(K) whose determinant cannot vanish if the trivial solution is implied.
It is of interest to consider briefly the propagating wave
solutions of the corresponding linear time-dependent problem,
T t ⫹w 0 T z ⫹u"“H T 0 ⫽0,
共91兲
so that w 0 ⫺c replaces w 0 in 共87兲, c being the phase velocity
of the wave. In this case, since K appears only in the single
condition 共89兲, det„A(„w 0 ( ␮ )⫺c…⫺ 1)…⫽0 is equivalent to
c⫽w 0 共 ␮ 兲
FIG. 5. Phase speed c vs ␮ for various wavenumbers n, 3D top-hat profile.
h 1 ⫽⫺U"“H T 0 ⫽U"ir ␦ 共 r⫺ ␮ 兲 .
共84兲
Thus we set ␾ 1 , ␺ 1 ⫽ f ( ␪ ,z,t)⌽ 1 (r), f ␪ ( ␪ ,z,t)⌿ 1 (r), where
f ⫽U"ir arises naturally from 共84兲. It is then seen that, for the
horizontal average over the cell we have
具 “ H ␾ 1 ⫹“ H Ãiz ␺ 1 典 ⫽ „⌽ 1 共 1 兲 ⫺⌿ 1 共 1 兲 …,
1
2
共85兲
and the boundary conditions must ensure that this vanishes.
The boundary conditions we suggest now will model the
situation where, as in the 2D periodic cases studied above,
each up cell is bordered by down cells, so that w changes
sign across boundaries. We impose
2
␺ n ⫽ⵜ H2 ␺ n ⫽0,
w⫽ ␾ n ⫽ ␺ n ⫽ⵜ H
r⫽1.
共86兲
For the top-hat profile, we have additional regularity condi2
tions. Taking the z derivative of ⫺ⵜ H
w⫽T, and using the
continuity and temperature equations we obtain the homogeneous equation,
4
␾ ⫹u"“H T 0 ⫽0.
w0 ⵜH
共87兲
4
␾ ⫽0, except at r⫽ ␮ , where the jump
We then have that ⵜ H
conditions
关 ␾ 兴 r⫽ ␮ ⫽ 关 ⳵ r ␾ 兴 r⫽ ␮ ⫽ 关 ⳵ r2 ␾ 兴 r⫽ ␮ ⫽0,
共88兲
and
K⫽w 0 共 ␮ 兲 ⫺1
关 ⳵ r3 ␾ 兴 r⫽ ␮ ⫽Ku r 共 ␮ 兲 ,
共89兲
prevail, 关 Q(r) 兴 r⫽ ␮ here denoting Q( ␮ ⫹)⫺Q( ␮ ⫺).
2
4
u)⫽0 we obtain ⵜ H
␺ ⫽0,
Now from “ H Ã(“ H p⫺ⵜ H
so with 共86兲 we have only the trivial solution ␺ ⫽0 in the
homogeneous case.
For ␾ we must satisfy 共87兲 and the six conditions from
4
␾ ⫽0 in separated
共86兲, 共88兲, 共89兲. From solutions of ⵜ H
2
form ␾ ⫽ f ⌽(r), f ␪␪ ⫽⫺n f we have, if n⭓1,
⌽⫽
再
C 1 r n ⫹C 2 r n⫹2 ,
C 3 r ⫹C 4 r
n
共 or r ln r
n⫹2
if 0⭐r⬍ ␮ ,
⫹C 5 r ⫺n ⫹C 6 r 2⫺n
if n⫽1 兲 ,
if ␮ ⭐r⬍1.
det„A 共 K 兲 …
.
det„A 共 0 兲 …
共92兲
Consequently, the condition det„A(K)…⫽0 that the quasisteady perturbations be uniquely determined by the mean
flow is the same as the condition that all traveling wave
solutions have nonvanishing wave speed, so that such waves
propagate to the boundaries of the domain. We show in Fig.
5 the wave speeds as a function of ␮ for several n, indicating
that the expansion is consistent for these terms, at least for a
substantial range of ␮.
Unfortunately a general proof of uniqueness using energy methods does not seem possible. From 共82兲, 共86兲, 共88兲,
taking ␺⫽0, we obtain
冕冕
r⬍1
2
␾ 兲 2 ⫹q ␾ 2 兴 dx dy⫽0,
关共 ⵜ H
冉
冊
2
“ Hⵜ H
w0
1
.
q⫽ “ H •
2
w0
共93兲
With conditions 共86兲 we have a Poincaré inequality
2
兰兰 (ⵜ H
␾ ) 2 dx dy⭓C 兰兰 ␾ 2 dx dy,C⬎0, so that 共93兲 implies
␾ ⫽0 provided q⬎⫺C for all r⬍1. The last inequality does
not generally hold for profiles of interest, as can be seen by
considering a smoothed approximation to the top hat, for
which q can have large positive and negative values in the
vicinity of the plume boundary.
We recall that the plumes shown in plate I are threedimensional. This picture reminds us that plumes originating
at one wall might extend only a small distance into the bulk
fluid before losing identity in the chaotic convective flow.
Also it is clear that, to take one case, near the lower wall we
expect to see mainly rising plumes. Various modifications of
the plume-in-cell model can be devised to reflect these properties of the flow. In calculations not presented here, we have
considered a model for a single ‘‘up’’ plume isolated in a
chaotic sea of plumes of both types. Here, a cell model must
account for the average ambient temperature in the vicinity
of the plume. In this case we impose the condition
dw/dr(1)⫽0 of zero vertical stress at the outer wall 共instead
of relying, as we have done above, on the equal numbers of
identical up- and down-plumes to cancel these stresses on
average兲. Conditions on ␺ , ␾ , corresponding to neighboring
plumes being of the same type, were then imposed. For this
problem we again obtained consistency of the problem in the
sense of this section, for a range of ␮ of the top-hat profile.
VII. DISCUSSION
共90兲
In general, with only a periodicity condition in x imposed in two dimensions, we have found that plumes of the
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38
Chaos, Vol. 10, No. 1, 2000
type considered in this paper are always vulnerable to a shift
instability. The latter produces a transient mean flow and a
horizontal displacement of the fluid. In addition, steady bifurcation analysis indicates that the plume is susceptible to a
second mode of instability, which mode carries a nonvanishing Eulerian mean flow in the steady case. This properties of
this mean-flow instability depend very much on the detailed
structure of the plume temperature profile.
In general, we can expect both modes to be present,
depending upon the boundary conditions imposed on vertical
walls. The evolution of plumes allowing both instabilities
could be studied by perturbations of codimension two. In the
discussion of Sec. V we chose instead to focus on the symmetric double-hat profile, so that a boundary-value problem
could be formulated which excluded the shift instability.
This allowed us to study the expansion in some detail and
derive an equation for the mean flow. This procedure, if restricted to the reduced system ␭⫽0, produces a solution of
the reduced equations 共14兲, and so we may apply Theorem I
to obtain a shifted solution of the reduced system, with mean
flow ␰ t (z,t)⫹U( ␶ ,z). In this case 共since ␭⫽0兲 there is no
linear cut-off at high wavenumber, and any saturation must
come from nonlinearity.
Although we have focused on the symmetric double-hat
plume, the plume-in-cell model might be applied to plumes
which do not exhibit top–bottom symmetry. The boundary
condition applied to u is only approximate, since it does not
follow from symmetry. Up- and down-moving plumes could
in this way be treated separately and then combined to obtain
an equation for the mean flow. In this case all of the terms in
Table II could appear. The lowest-order linear term is then of
the form DU. It can be shown that this term can reduce the
␦ needed for the mean-flow instability. This suggests that the
non-Boussinesq effects introduced in Refs. 11,12 might partially account for the ease with which mean flows were established in that experiment.
The ideas developed in the two-dimensional case carry
over to three dimensions, where the plume geometry is
richer, and there is an analogous cylindrical plume-in-cell
model for analyzing the mean-flow instability.
We have not considered time-periodic eigenfunctions of
the linear problem. In the experiments reported in Refs.
11,12 the mean flow is observed to be almost steady and the
plume formation is an irregular process 共in space and time兲.
Time-dependence of the plume perturbations can create Lagrangian mean flow, with mean horizontal migrations of
Stephen Childress
fluid particles. We have studied here only the Eulerian mean
flow, and shown that for slender plumes how it can arise
from an instability distinct from the shift instability.
When TB symmetry is broken, the leading O( ⑀ ␦ ) contribution to the Reynolds stress appears to be similar in some
respects to the anisotropic kinetic alpha 共AKA兲 effect first
studied by Frisch, She, and Sulem;15 see also Ref. 16. The
AKA instability was studied in the context of forced Navier–
Stokes flows which lack parity invariance, meaning a lack of
invariance to reflection in both velocity and coordinate vectors. Applied to Bénard convection, a reflection in temperature also occurs, and the TB invariance is then a particular
case of parity invariance. The AKA mean flow is also driven
by a force proportional to mean shear. On the other hand the
AKA theory is a first-order theory, whereas we have obtained instability at second order in flows possessing TB
symmetry.
ACKNOWLEDGMENTS
This research was supported at the Courant Institute of
Mathematical Sciences, New York University, by the National Science Foundation under Grant No. DMS 970-4546.
Part of the work was done at the University of Exeter on a
PPARC Visiting Fellowship, Grant No. GR/L20191. The author is thankful to the staff of the Department of Mathematics for their hospitality and encouragement during this visit.
Thanks to Andrew Soward and Andrew Gilbert for pointing
out the connection of this work with the AKA instability.
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